Eventual joint fractional certificate and guarded signed-unit bank states

Let $c>C_0$, $h_n=\lceil cn/\log n\rceil$ and $M_n=2n+h_n$. For all sufficiently large $n$, there is a joint guarded central anchor, charged fractional selector, and family of literal two-state bank slots satisfying the accompanying definition.

At every prime $p\le y$ there are $\beta_p$ slots of each sign, with

Each slot consists of two finite sets of legal positive factors, whose product-valuation difference is exactly $+e_p$ or $-e_p$. Both states avoid the anchor, fixed factors, and all candidates; states belonging to different slots are pairwise disjoint. The charged initial bank is precisely the union of all zero states.

The fractional weights lie in $[0,1]$, have integer sums on complete rough-signature rows, and satisfy the exact charged valuation balance. The anchor and divisor $D$ obey

This is the remaining geometric and analytic existence obligation. Its hypotheses concern individual factor states and their joint placement; universality for every bounded error follows separately from the finite signed-bank theorem.